Y-type Flux-Tube Formation in Baryons

0.33 MB·

by  Toru T. Takahashi

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Y-TYPE FLUX-TUBE FORMATION IN BARYONS
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n
a TORU T. TAKAHASHI
J
Yukawa Institute for Theoretical Physics,
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Kyoto University, Sakyo, Kyoto 606-8502, Japan

E-mail: [email protected]
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v
1 HIDEOSUGANUMAANDHIROKOICHIE
0
Faculty of Science, Tokyo Institute of Technology,
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1 Ohokayama 2-12-1, Tokyo 152-8551, Japan
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0 For morethan 300 different patterns of the 3Q systems, the ground-state 3Q po-
/ tential Vg.s. is investigated using SU(3) lattice QCD with 123 ×24 at β = 5.7
t 3Q
a and 163×32 at β = 5.8,6.0 at the quenched level. As a result of the detailed
l analyses, we find that the ground-state potential Vg.s. is well described with so-
- 3Q
ep bcaeltlteedrYth-aannsa1t%z.asHVe3rQe,=Lm−inA3dQenPotie<sjt|hrei−1mrjin|+imσa3lQvLalmuien+ofCt3oQta,lwfliuthx-tthuebeaclceunrgatchy.
h We also study the excited-state potential Ve.s. using lattice QCD with 163×32
: 3Q
v at β = 5.8,6.0 for more than 100 patterns of the 3Q systems. The energy gap
Xi between V3gQ.s. and V3eQ.s., which physically means the gluonic excitation energy,
is found to be about 1 GeV in the typical hadronic scale. Finally, we suggest a
r possiblescenariowhichconnects thesuccess ofthequarkmodeltoQCD.
a
1. Introduction
Itis widelyacceptedthatQuantumChromodynamics(QCD)isthe under-
lying theory of the strong interaction for hadrons and nuclei. In a spirit of
the particle physics, it is strongly desired to describe the hadron dynamics
directly from QCD. However, due to the strong coupling nature of QCD
in the infrared region, it still remains a big challenge to derive physical
quantities directly from QCD in an analytic manner.
For the nonperturbative analysis of QCD, lattice QCD Monte Carlo
calculations have been established as a reliable method directly based on
QCD. Lattice QCD calculations enable us to perform a nonperturbative
quantitative analysis in a model-independent way. For instance, lattice
QCD calculations have been very successful in reproducing hadron mass
spectra.1 Nevertheless,theseresultshardlyinformustheinternalstructure
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ofhadrons. It wouldbe alsoindispensable to comprehendhadronsdirectly
from quarks and gluons, the degrees of freedom of QCD.
For this aim, we cast light on the inter-quark potentials, which are
directly responsible for the hadron properties and their internal structures
at the quark-gluon level.
2. Ground-state three quark potential
For the quark-antiquark (QQ¯) potential, which is one of the most funda-
mentalquantitiesinQCD,manylatticeQCDstudieshavebeenperformed.
As a result, QQ¯ potential is known to be reproduced by a simple sum as
VQQ¯ =−AQrQ¯ +σQQ¯r+CQQ¯ with r the distance between quark and anti-
quark. Here, σ ≃ 0.89 GeV/fm denotes the string tension in the flux-tube
picture.
However, in contrast with a number of lattice QCD studies for the
QQ¯ potential, almost no lattice studies were done before our study2,3 for
the three-quark (3Q) potential V , which is directly responsible for the
3Q
baryon properties. We study the ground-state 3Q potential on more than
300 patterns of the spatial quark configurations, using SU(3) lattice QCD
at the quenched level with 123 × 24 at β = 5.7 and with 163 × 32 at
β =5.8,6.0.2,3
As a result, we find that the 3Q potential V can be reproduced by
3Q
so-called Y-ansatz as 2,4
1
V3Q =−A3QX|r −r | +σ3QLmin+C3Q, (1)
i j
i<j
with the accuracy better than 1% for all the different patterns of quark
configurations. Here, L denotes the minimal value of total flux-tube
min
length,5,6 which is schematically illustrated in Fig.1.
In the fit analysis with Y-ansatz, we find two remarkable features2; the
universalityofthestringtensionasσ3Q ≃σQQ¯ andtheone-gluon-exchange
consequence as A3Q ≃ 21AQQ¯.
A
c b
120 120
P
L mi n=-AP+-BP+C-P
B
a C
Figure1. Theflux-tubeconfigurationinthe3Qsystemwiththeminimallengthofthe
Y-typeflux-tube, Lmin.
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Furthermore, we consider also Y-ansatz with Yukawa-type two-body
forceasVYukawa =−AYukawa exp(−mB|ri−rj|)+σYukawaL +CYukawa,
3Q 3Q Pi<j |ri−rj| 3Q min 3Q
which is a conjecture from the dual superconductor scenario on the QCD
vacuum. Here, m denotes the dual gluon mass in the dual superconduc-
B
tor picture. However, we find no evidence of the Yukawa-type two-body
force2 andagainconfirmthe adequacyof the Y-ansatzformin Eq.(1),i.e.,
σYukawa ≃σ , AYukawa ≃A , m ≃0.
3Q 3Q 3Q 3Q B
Finally,asadirectevidenceofthe Y-typeflux-tubeformation,weshow
inFig.2theflux-tubeprofileinthespatially-fixed3Qsystemobtainedusing
maximally-abelian(MA)projectedlatticeQCD.7 TheactiondensityinR3
is calculated under the presence of spatially-fixed three quarks in a color-
singlet state. Here, the distance between the junction and each quark is
about0.5 fm. We thus observeY-type flux-tube formationinlattice QCD.
20
18
16
14
12
10
8
6
17.5 20 22.5 25 27.5 30 32.5
Figure 2. The flux-tube profile in the spatially-fixed 3Q system, inthe MA projected
QCD.6 Thedistancebetween thejunctionandeachquarkisabout0.5fm.
3. Excited-state three quark potential
3.1. Lattice QCD study of gluonic excitations
We also investigate the gluonic excitation in the spatially-fixed 3Q system
using SU(3) lattice QCD. In the flux-tube picture,5,6 gluonic excitation
modes would be regarded as the vibrational modes of a flux-tube. In the
valencepicture,theycorrespondtothehybridhadronssuchasqq¯GorqqqG.
Inparticular,hybridhadronsareprobablecandidatesoftheexotichadrons,
which has exotic quantum numbers such as JPC =0−−,0+−,1−+,2+−,...,
whichcannotbeconstructedwithinasimplequarkmodel. Itisthenworth
investigatinggluonicexcitationsdirectlyfromQCD,fromtheexperimental
viewpoint as well as the theoretical viewpoint.8
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In Fig.3 we show the first lattice QCD results for the excited-state 3Q
potential obtained with 163 ×32 lattice at β = 5.8,6.0 at the quenched
level.9 Here, the horizontal axis denotes the minimal length of the Y-type
flux-tube, L , as a label to distinguish the three-quark configuration.
min
The vertical axis denotes the energy induced by three static quarks in a
color-singlet state. The open symbols are for the ground-state potential
Vg.s. discussed in the previous section, and the filled symbols are for the
3Q
excited-state potential Ve.s.in the 3Q system. The energydifference ∆E ≡
3Q
Ve.s.−Vg.s. physically means the gluonic excitation energy.
3Q 3Q
)4 )5
8 0
5. 6.
= =
β β
( (4
V] 3 V]
e e
G G
[ [3
al al
ti2 ti
n n
e e2
t G.S. t G.S.
o o
p 1st E.S. p 1st E.S.
Q Q
31 31
0 0.5 1 1.5 0 0.5 1 1.5
L [fm] L [fm]
min min
Figure3. ThelatticeQCDresultsfortheground-state3QpotentialVg.s.(opencircles)
3Q
andthe1stexcited-state3QpotentialV3eQ.s.(filledcircles)asthefunctionofLmin. These
latticeresultsatβ=5.8andβ=6.0wellcoincidebesidesanoverallirrelevantconstant.
We find that the gluonic excitation energy ∆E is more than 1 GeV in
the typical hadronic scale as 0.5fm≤L ≤1.5fm, which is large in com-
min
parisonwiththeexcitationenergiesofquarkorigin. (FortheQQ¯ system,a
largegluonicexcitationenergyisreportedinrecentlattice studies.10)Such
a gluonic excitation would contribute significantly in the highly-excited
baryonswith the excitationenergyabove 1 GeV. in fact, the lowesthybrid
baryon,8 which is described as qqqG in the valence picture, is expected to
havealargemassofabout2GeV.Thislargegluonicexcitationenergyalso
gives us the reason for the great success of the simple quark model.9
3.2. Success of the quark model
The simple quark model contains only constituent quarks as explicit de-
greesoffreedom,anddoesnothavegluonicmodes. Nevertheless,thequark
model has been very successful in reproducing the low-lying hadron prop-
erties and spectra especially for baryons, in spite of the non-relativistic
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treatment and the absence of gluonic excitation modes. It is true that
the non-relativistic treatment for hadron mass spectra can be justified by
the spontaneous chiralsymmetry breaking,which gives rise to a large con-
stituent quark mass of about 300 MeV, but we have no reason based on
QCD for the absence of the gluonic mode. Now, directly based on QCD,
the gluonic excitation energy ∆E is found to be large, in comparison with
quark excitation energy. We propose a possible scenario connecting the
success of the quark model to QCD3,9 in Fig.4.
Figure4. ApossiblescenarioconnectingthesuccessofthesimplequarkmodeltoQCD.
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